Lemma 4.7 ([1]}, Lemma 2.4; see also [2]}, Lemma 3.7)  
Let \(\sigma \in \mathbb {N}\)  and \(f,g:\mathbb {T}^d\rightarrow {\mathbb {R}}\)  be smooth functions. Then for every \(p\in [1,+{\infty }]\) ,
\(\big |\Vert fg(\sigma \cdot )\Vert _{L^p(d)}-\Vert f\Vert _{L^p(d)}\Vert g\Vert _{L^p(d)} \big |\lesssim \sigma ^{-\frac{1}{p}}\Vert f\Vert _{C^1(d)}\Vert g\Vert _{L^p(d)}.\) 

